Absence of edge reconstruction for quantum Hall edge channels in graphene devices

Quantum Hall (QH) edge channels propagating along the periphery of two-dimensional (2D) electron gases under perpendicular magnetic field are a major paradigm in physics. However, groundbreaking experiments that could use them in graphene are hampered by the conjecture that QH edge channels undergo a reconstruction with additional nontopological upstream modes. By performing scanning tunneling spectroscopy up to the edge of a graphene flake on hexagonal boron nitride, we show that QH edge channels are confined to a few magnetic lengths at the crystal edges. This implies that they are ideal 1D chiral channels defined by boundary conditions of vanishing electronic wave functions at the crystal edges, hence free of electrostatic reconstruction. We further evidence a uniform charge carrier density at the edges, incompatible with the existence of upstream modes. This work has profound implications for electron and heat transport experiments in graphene-based systems and other 2D crystalline materials.


I. SAMPLE DETAILS AND AFM MAPPING
The sample AC04 studied is this work is a heterostructure made of a graphene sheet atop a hexagonal boron nitride (hBN) flake, assembled by van der Waals stacking, and then deposited on a p ++ Si/SiO 2 substrate to enable back gating of the charge carrier density in graphene. The voltage bias V b is applied using a Cr/Pt/Au contact patterned by e-beam lithography and covering partially the graphene sheet, leaving a large fraction of the perimeter accessible by the tip for imaging and tunneling spectroscopy of the edge states, see Fig. S1A and B. The graphene bulk properties of this sample have been presented in Ref. [30].
The STM tip is brought atop the graphene sheet by AFM imaging of the coding markerfield patterned on the whole chip surface. This guiding process is done after about ten AFM images.
An AFM mapping of graphene and its boundary with the underlying hBN performed at B = 14 T is shown in Fig. S1C. High-resolution AFM images of some edges are placed in overlay. These images reveal that the vacuum annealing employed to clean the graphene left some resist residues that have migrated toward the edges, forming bright spots in-between which edges are clean. In this work we focus on the edge indicated by the white arrow, which is also the direction of the Current Imaging Tunneling Spectroscopy (CITS) measurement grids performed from the bulk of graphene to the edge. Note that the tuning fork we used here still displays a relatively high quality factor in magnetic field, with Q ∼ 4000 at 14 T (Fig. S1D).

II. LOCALIZATION OF GRAPHENE EDGES ON hBN
We show in   We believe that the instability of the tunneling current measured on the very edge of the graphene in Fig. S2 stems from the local lifting of the graphene sheet edge from the hBN flake, each time the STM tip scans over it, due to electrostatic interactions with the tip.

III. TIP-INDUCED LIFTING OF THE GRAPHENE EDGE AND DEFINITION OF THE EDGE POSITION
We discuss here another way to locate the edge by means of a CITS grid spectroscopy measurement of the spatial dispersion of the Landau level (LL) spectrum toward the boundary. The grid spectroscopy is set to start far away in graphene bulk and to finish a few nanometers beyond the edge, previously located with STM images. Moreover, the slow x-axis direction of the grid is chosen to be perpendicular to the edge. A safety condition is added to the Z-controller to prevent the tip from crashing into hBN : if the Z-position of the tip goes below a threshold (typically 3 nm below the Z-position of the tip estimated close to the edge), the tip is withdrawn and the CITS ends.
We show in Fig. S3A the topographic map z(d edge , y) obtained from a CITS toward the graphene armchair edge identified in Fig. S2. d edge is the distance from the armchair edge, while y is the lateral coordinate parallel to the edge. The topographic map features a clean and flat bulk graphene on a 80 × 10 nm 2 area next to the edge. When the tip is situated a few nanometers away from the edge, the z(d edge , y) map reveals inhomogeneous bright spots. Though one can first think about residues, the small height of these spots, around 1 − 3Å, rules out this hypothesis.
We rather attribute these large spots to the lifting of the edge of the graphene sheet, as illustrated in Fig. S3D. The attractive van der Waals force of the tip was shown [33] to lift locally a graphene sheet lying on a SiO 2 substrate on a typical height of 1Å. Although we do not observe such lifting in Fig. S3A in bulk graphene (either because the deformation follows the tip such that we eventually observe an overall flat background, or because the deformation of the graphene sheet on hBN is more difficult, since the adhesion interactions between both materials are more important than between graphene and SiO 2 ), we can assume that the graphene flake is more easily deformed at the edge by the force of the tip, and therefore the lifting is larger there than in the bulk.
The lifting of the edge is well visible in the height profile of Fig. S3B, obtained by averaging the topographic map along the y direction (parallel to the edge). The z profile features a flat region corresponding to bulk graphene (with variations of less than 1Å), and a hump of 3Å height at the edge. After that, the tip quickly moves down by several nanometers until it meets the safety condition of the Z-controller, which stops the CITS. We attribute this lowering of the z position to the fact that the tip apex has gone beyond the edge of graphene, but tunneling remains possible with some other higher atoms of the tip close to the apex, see Fig. S3D. This makes the measurement of a tunneling current possible even when the apex itself is lying on hBN, yet this current is highly unstable.
From this model we assume the position of the edge of graphene (i.e. the tip apex is atop the edge) is given by the maximum of the hump in the z profile, and from this origin we compute d edge the distance from the edge, which we use in the main text and the following figures.

IV. ADDITIONAL TUNNELING CONDUCTANCE MAPS AT THE EDGE
We show in this section two additional tunneling conductance maps acquired along the same armchair edge, but a few tens of nanometers away from the map shown in Fig. 2 of the main text.
The back-gate voltage is fixed at V g = −5 V, corresponding to filling factor ν = 0. We now consider in more details the tunneling conductance map shown in Fig. S4B. We plot in Fig. S6A the evolution of the positions in energy E N of the visible LL N peaks and in Fig. S6B the variation of their height as a function of d edge . The amplitude of the peaks decreases as we approach the edge until peaks merge into a V-shape background at the edge where they are no longer visible. In particular, LL 4 vanishes at 9 l B from the edge, LL 3 at 3 l B whereas LL 2 and LL ±1 disappear at l B . The amplitude of LL 1 also vanishes way faster than the other LL N of higher index N . In addition to the peak vanishing at the edge, we can also notice in Fig. S4B and S6A a weak dispersion toward higher energy of the LL N peaks close to the edge (on a length of around ∼ 6 l B from the edge), see Ref. [27].
The bulk value v * F,bulk = 1.42 × 10 6 m.s −1 is consistent with a renormalization of the Fermi velocity due to the enhancement of electron-electron interactions at charge neutrality [47,48,66], as characterized in a previous work [30] for the same sample AC04. Below 7 l B the effective Fermi velocity starts to increase toward the armchair edge due to the dispersion of the LL peaks, reaching v * F,edge = 1.6 × 10 6 m.s −1 at l B from the edge. As for the carrier density, we obtain a residual value n 0 ≈ 7 × 10 9 cm −2 in bulk graphene (in agreement with a back-gate voltage tuned at ν = 0). Below 60 nm the density is seen to decrease and eventually vanishes at 40 nm = 5 l B from the edge. A similar decrease of the density with respect to its bulk value has also been observed around l B from graphene edge on graphite [27]. Finally, we use the v * F (d edge ) and E D (d edge ) parameters to plot in Fig. S6A the fitted energies of each LL (black dashed lines). We notice a good agreement with the experimental points, especially for the dispersing parts.

THE EDGE
We show in this section additional tunneling conductance gate maps (Fig. S7)  to be in the middle of this range, where the ν = 0 gap is maximal, see white arrow. In panels (a-h), only the LL 0− peak is well visible, the LL 0 + peak is hindered. In panel (A), the kink at charge-neutrality is hardly visible for LL 0 , we rather identify it in the charging peaks below it.